# Consistency constraints for overlapping data clustering

@article{Culbertson2016ConsistencyCF, title={Consistency constraints for overlapping data clustering}, author={Jared Culbertson and Dan P. Guralnik and Jakob Hansen and Peter F. Stiller}, journal={ArXiv}, year={2016}, volume={abs/1608.04331} }

We examine overlapping clustering schemes with functorial constraints, in the spirit of Carlsson--Memoli. This avoids issues arising from the chaining required by partition-based methods. Our principal result shows that any clustering functor is naturally constrained to refine single-linkage clusters and be refined by maximal-linkage clusters. We work in the context of metric spaces with non-expansive maps, which is appropriate for modeling data processing which does not increase informationâ€¦Â

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## References

SHOWING 1-10 OF 28 REFERENCES

### An Impossibility Theorem for Clustering

- Computer ScienceNIPS
- 2002

A formal perspective on the difficulty in finding a unified framework for reasoning about clustering at a technical level is suggested, in the form of an impossibility theorem: for a set of three simple properties, it is shown that there is no clustering function satisfying all three.

### Characterization, Stability and Convergence of Hierarchical Clustering Methods

- MathematicsJ. Mach. Learn. Res.
- 2010

It is shown that within this framework, one can prove a theorem analogous to one of Kleinberg (2002), in which one obtains an existence and uniqueness theorem instead of a non-existence result.

### Enhanced Topology-Sensitive Clustering by Reeb Graph Shattering

- Computer Science
- 2011

Preliminary experimental results are provided to demonstrate that the improved topology-sensitive clustering algorithm yields a more accurate and reliable description of the topology of the underlying scalar function.

### One-to-One Correspondence Between Indexed Cluster Structures and Weakly Indexed Closed Cluster Structures

- Mathematics, Economics
- 2007

We place ourselves in a setting where singletons are not all required to be clusters, and we show that the resulting cluster structures and their corresponding closure under finite nonemptyâ€¦

### Classifying Clustering Schemes

- Computer ScienceFound. Comput. Math.
- 2013

A framework is constructed for studying what happens when one imposes various structural conditions on the clustering schemes, under the general heading of functoriality, and it is shown that, within this framework, one can prove a theorem analogous to one of Kleinberg (Becker etÂ al).

### Persistence-Based Clustering in Riemannian Manifolds

- Computer ScienceJACM
- 2013

A clustering scheme that combines a mode-seeking phase with a cluster merging phase in the corresponding density map, and whose output clusters have the property that their spatial locations are bound to the ones of the basins of attraction of the peaks of the density.

### Combinatorial optimisation and hierarchical classifications

- Computer ScienceAnn. Oper. Res.
- 2007

Within the galaxy of optimization, some selected topics relating Combinatorial Optimization and Hierarchical Classification are discussed, including NP-completeness results and search for polynomial instances, and some standard algorithmic approaches are discussed.

### The Construction of Hierarchic and Non-Hierarchic Classifications

- Computer ScienceComput. J.
- 1968

A theoretical framework within which the properties of cluster methods, which operate on data in the form of a dissimilarity coefficient on a set of objects, may be discussed is outlined.

### Weak Hierarchies: A Central Clustering Structure

- Mathematics
- 2014

The k-weak hierarchies, for k â‰Ą 2, are the cluster collections such that the intersection of any (k + 1) members equals the intersection of some k of them. Any cluster collection turns out to be aâ€¦